Infinite Series

• Series of Positive Terms

We shall denoting by , the set of all such series whose terms are positive. Thus, is series of positive terms if and only if . If then its sequence or partial sums is monotonically increasing for . This gives Theorem: If then either converges to positive number or diverges to . Proof: If and […]

• Decimal Representation in Series

Consider the series Where and are integers lying between 0 and 9. From monotonically non-decreasing bounded partial sums, if follows that (*) is convergent. The series (*) is often abbreviated as (called decimal representation) and is used for the real number which is the sum of the series (*). Thus every decimal representation represents a […]

• Convergence of an Infinite Series

An infinite series is said to converge, diverge or oscillate according as its sequence of partial sums converges, diverges or oscillates. In case converges to s, then s is called the sum of the series and we shall write or In this series shall be written as Thus the infinite series shall be denoted by […]

• Meaning of an Infinite Series

If be given real valued sequence, then an expression of the form is called an infinite series. In symbols it is generally written as or . If all the terms of after a certain number are zero then the expression , written is called a finite series. Simply speaking, even without any reference to the […]

• Introduction to Infinite Series

A.D. that the wider significance of finite and infinite series was realized. The finite series generally do not involve any difficulty in respect of the validity of application of algebraic operations thereto, as compared to the infinite series. The application of algebraic operations to infinite series requires the additional concept of the convergence of series. […]